SAPZ Singularity Principle for the 3D Navier--Stokes Equations: A Spectral--Entropy Threshold Criterion with Route--T Discharge (v5.5r2)

# Overview This record releases **v5.5r2** of a two-paper set developing the SAPZ (Spectral--Averaged Parabolic Zone) principle for the 3D incompressible Navier--Stokes equations. **Files included in this record (PDF only):**- Main paper (PDF): `SAPZ_Singularity_Principle_Navier-Stokes_v5.5r2.pdf`- Companion paper (PDF): `Aux_Proof_v5.5r2.pdf` The framework is organized around a verifiable, scale-uniform threshold envelope built from the convolution-first energy-density observable\[δ_ε(t):= \|\, |∇ u(·,t)|^2 * φ_ε \,\|L^∞_x,δ(t) := 0<ε≤ ε_0δ_ε(t),\]together with a universal threshold level \(δ_c>0\) determined by fixed analytic profiles (mollifier / cutoffs / normalization) and the viscosity. # Main statements (high-level) ## Continuation criterion (finite horizon) For an energy-class weak solution \(u\) in the Leray--Hopf framework, the main paper establishes a continuation criterion of the following form: - If a uniform-scale subcriticality bound below the universal threshold holds on a given horizon \((0,T)\), then \(u\) is smooth on \((0,T]\) and continues beyond \(T\).- Conversely, any finite-time singularity forces threshold reach in the quantitative necessity sense formulated in the main paper. ## Companion closure interface (Route--T / Gate A / Gate B) The companion paper supplies theorem-level modules implementing the proof interface: - **Gate A:** approximate-identity \(L^∞\) identification (a Lebesgue differentiation/approximate-identity mechanism applied to \(f(·,t)=|∇ u(·,t)|^2\)).- **Route--T (transport-bypass extraction):** defect \(⇒\) strictly positive transport residual, using high-frequency filtering and commutator extraction with explicit error budgets.- **Gate B:** standard CKN \(ε\)-regularity closure. # What is new in v5.5r2 (this record) v5.5r2 is a **reviewer-defense and interface-consistency release** focused on eliminating “instant reject triggers”: 1) **Cross-document theorem-number consistency.** The main paper’s proof-input references are synchronized with the companion’s numbering for the kinematic CKN-bypass module and the optional legacy robustness module. 2) **Object dictionary for residual terms (to prevent scale/notation misreadings).** The companion inserts an explicit one-page dictionary separating (i) the CT2(T) energy-level remainder/envelope objects from (ii) the Route--T transport residual and the defect functional \( T_ε(t)\), preventing the common misreading in which an \(O(ε)\) upper bound is mistakenly applied to the localized residual used in the Route--T positivity extraction. 3) **Explicit CKN trigger definition in the main paper.** The main paper now fixes the precise CKN-type concentration quantity used in the kinematic bypass step: \[ C(u;Q_r) := r⁻¹Q_r|∇ u|^2, \] avoiding confusion with other standard \(ε\)-regularity formulations (e.g. involving \(u\) and \(p\)). # Scope The framework targets standard 3D incompressible Navier--Stokes settings, including whole-space/periodic geometries and bounded no-slip domains (via boundary-normalized variants handled in the companion). The writing is modular: the main paper isolates the criterion statement, while the companion isolates analytic modules and proof interfaces. # Recommended citation Lee Byoungwoo, "SAPZ Singularity Principle for the 3D Navier--Stokes Equations: A Spectral--Entropy Threshold Criterion with Route--T Discharge" (Version v5.5r2), with companion "Auxiliary Proof Modules for the SAPZ Singularity Principle" (Version v5.5r2), Zenodo, 2026.